This document discusses the single sample t-test, which is used to determine if the mean of a sample is statistically significantly different from the population mean. It explains how to calculate the t-statistic and compare it to the critical t-value based on the confidence level and degrees of freedom. Examples are provided to demonstrate comparing sample means to population means and determining whether to reject or fail to reject the null hypothesis that the means are equal. Factors that influence the t-calculation are also outlined.

1. Unit 4: Statistical Significance Lesson 2: Mean Differences - The Single Sample t-test EDER 6010: Statistics for Educational Research Dr. J. Kyle Roberts University of North Texas Next Slide

2. Mean Differences IQ Scores Next Slide 100 115 85 μ IQ = 100 σ IQ = 15

3. The Single Sample t-test “Does my sample mean differ statistically from the population mean?” “Are the sample and population means statistically significantly different from each other?” CI 95% = 105 ± 6.57 CI 95% = 105 ± 1.31 Next Slide N = 20 SD = 15 Mean = 100 Mean = 105 Population Sample Study 1 N = 20 SD = 3 Mean = 100 Mean = 105 Population Sample Study 2

4. Computing the ‘t’ statistic Range from - ∞ to +∞ What does a t = 0 mean? Next Slide

5. Alpha and the Null Hypothesis “ α = the point at which we reject our null hypothesis” 1 – α = Confidence Interval The probability of rejecting a Null Hypothesis when it is true CI 95% : α = .05 Next Slide

6. Critical Values for the t Distribution 95% Fail to Reject Area Next Slide -1.96 SD +1.96 SD 5% Reject Area 5% Reject Area

7. Critical Values for the t Distribution μ IQ = 100 σ IQ = 15 Next Slide

8. t Distribution Chart df = n-1 Level of significance = α t-critical for our study = 2.110 Page 637 in Hinkle et al. 4 th ed Page 638 in 5 th ed Next Slide

9. Determining Statistical Significance -2.110 t-critical 2.110 t = 4.72 We reject the hypothesis that our sample mean equals the population mean Next Slide -1.96 SD +1.96 SD

10. Comparing Studies t-crit = 2.093 Fail to reject the Null Hypothesis Reject the Null Hypothesis Next Slide N = 20 SD = 15 Mean = 100 Mean = 105 Population Sample Study 1 N = 20 SD = 3 Mean = 100 Mean = 105 Population Sample Study 2 CI 95% = 105 ± 6.57 SEM = 3.35 t-calc = 1.49 CI 95% = 105 ± 1.31 SEM = 0.67 t-calc = 7.46

11. Comparing Studies (again) CI 95% = 105 ± 6.57 CI 95% = 105 ± 0.89 SEM = 3.35 t-calc = 1.49 SEM = 1.36 t-calc = 3.68 t-crit = 2.093 Fail to reject the Null Hypothesis Reject the Null Hypothesis t-crit = 1.980 Next Slide N = 20 SD = 15 Mean = 100 Mean = 105 Population Sample Study 1 N = 121 SD = 15 Mean = 100 Mean = 105 Population Sample Study 2

12. Comparing Studies (one last time) CI 95% = 105 ± 6.57 CI 95% = 115 ± 6.57 SEM = 3.35 t-calc = 1.49 SEM = 3.35 t-calc = 4.48 Fail to reject the Null Hypothesis Reject the Null Hypothesis t-crit = 2.093 Next Slide N = 20 SD = 15 Mean = 100 Mean = 105 Population Sample Study 1 N = 20 SD = 15 Mean = 100 Mean = 115 Population Sample Study 2

13. Factors Affecting t-calc 1. The magnitude of the difference between the two means 2. The “spreadoutness” of the scores 3. The number of people in the study 4. The level at which α is set Next Slide

14. Running a Single Sample t-test in SPSS Analyze  Compare Means  One-Sample T Test Next Slide

15. Running a Single Sample t-test in SPSS (cont.) t-crit=2.365 Next Slide

16. Determining Statistical Significance by Alpha α is also: p-critical If: p-calc < p-crit, Reject the H 0 Next Slide α = .05 -2.365 t-critical 2.365 α = .05 p-calc

17. Determining Statistical Significance Means are not different statistically Fail to Reject the Null Hypothesis t-calc ≤ t-crit p-calc ≥ p-crit Means are different statistically Reject the Null Hypothesis t-calc > t-crit p-calc < p-crit Next Slide Statistically Significant Not Statistically Significant

18. Unit 4: Statistical Significance Lesson 2: Mean Differences - The Single Sample t-test EDER 6010: Statistics for Educational Research Dr. J. Kyle Roberts University of North Texas